# Dedekind domain

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A Dedekind domain is a Noetherian domain ${\displaystyle o}$, integrally closed in its field of fractions, so that every prime ideal is maximal.

These axioms are sufficient for ensuring that every ideal of ${\displaystyle o}$ that is not ${\displaystyle (0)}$ or ${\displaystyle (1)}$ can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of ${\displaystyle o}$.

This product extends to the set of fractional ideals of the field ${\displaystyle K=Frac(o)}$ (i.e., the nonzero finitely generated ${\displaystyle o}$-submodules of ${\displaystyle K}$).

## Useful properties

1. Every principal ideal domain is a unique factorization domain, but the converse is not true in general. However, these notions are equivalent for Dedekind domains; that is, a Dedekind domain ${\displaystyle A}$ is a principal ideal domain if and only if it is a unique factorization domain.
2. The localization of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a field or a discrete valuation ring.

## Examples

1. The ring ${\displaystyle \mathbb {Z} }$ is a Dedekind domain.
2. Let ${\displaystyle K}$ be an algebraic number field. Then the integral closure ${\displaystyle o_{K}}$of ${\displaystyle \mathbb {Z} }$ in ${\displaystyle K}$ is again a Dedekind domain. In fact, if ${\displaystyle o}$ is a Dedekind domain with field of fractions ${\displaystyle K}$, and ${\displaystyle L/K}$ is a finite extension of ${\displaystyle K}$ and ${\displaystyle O}$ is the integral closure of ${\displaystyle o}$ in ${\displaystyle L}$, then ${\displaystyle O}$ is again a Dedekind domain.